An Algebra Problem from RMO

Algebra Level 3

{f(x)=x3+ax2+bx+cg(x)=x3+bx2+cx+a\begin{cases} f(x)={ x }^{ 3 }+{ a }x^{ 2 }+bx+c \\ g(x)={ x }^{ 3 }+{ b }x^{ 2 }+cx+a \end{cases} Given that a,b,ca, b, c are integers with c0c\neq 0 and the following conditions hold for the above functions:

(a) f(1)=0f(1) = 0;

(b) the roots of g(x)=0g(x) = 0 are the squares of the roots of f(x)=0f(x) = 0.

Find the absolute value of (a2013+b2013+c2013).({ a }^{ 2013 }+{ b }^{ 2013 }+{ c }^{ 2013 }).

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